Graphics Reference
In-Depth Information
to-one. For example, natural parameterizations of a cylinder or sphere (see the exam-
ples below) are only regular and/or one-to-one on the interior of their domains.
However, the definition of volume and the formulas for computing it easily extend in
such special cases because they have the property that the boundary of the domain
is a set of measure 0 and the parameterization is regular and one-to-one on the inte-
rior of the domain. The proof would involve a straightforward limit process. We shall
use such extended formulas in the examples below. The reader who is uncomfortable
with this should compute areas for slightly smaller regions in the surface. Finally,
although we will not do this here, it should be pointed out that one
can
also define
the volume of an
arbitrary
n-dimensional submanifold of
R
n+1
. If there is no single
parameterizing function, then the definition is based on piecing together the volumes
defined for nice local parameterizations like in the case for length.
9.8.10. Example.
To find the area of the bounded cylinder
{
}
C
=
(
)
2
2
x y z
,,
x
+=
y
10 1
and
££
z
,
Solution.
Using the parameterization
(
)
=
(
)
(
)
Œ
[
]
¥
[]
F uv
,
cos u, sin u, v
,
uv
,
0 2
,
p
0 1
,
,
and the computations made in Example 9.8.3, we see that g = 1 and by Theorem 9.8.8
area
()
=
ÚÚ
12
=
p
.
[
]
¥
[]
02
,
p
01
,
9.8.11. Example.
To find the area of the sphere
S
of radius r about the origin.
Solution.
We again use a spherical coordinate parameterization
(
)
=
(
)
(
)
Œ
[
]
¥
[
]
F qf
,
r
cos
f
sin ,
f
r
sin
q
sin ,
f
r
cos
f
,
qf
,
0 2
,
p
0
,
p
.
Now, Example 9.8.4 showed that E = r
2
sin
2
f, F = 0, G = r
2
. Therefore,
2
4
2
EG
-=
F
r
sin f
and
()
=
ÚÚ
2
2
area
r
sin
fp
=
4
r
.
[
]
¥
[
]
02
,
pp
0
,
We finish this section with some comments on the extent to which the results
above are influenced by the parameterization that was chosen.
First of all, one could define the first fundamental form (or Riemannian metric)
by means of a parameterization and equation (9.37). It will be a good exercise in
the use of the chain rule and the definition of tangent vectors to show directly that
such a definition would be independent of the parameterizations. Let
U
,
V
Õ
R
n
and
let
n
n
Y
:
VM
Æ
and
F
:
UM
Æ
